Congruent Dudeney dissections of polygons: All the hinge points on vertices of the polygon

Abstract

Let α and β be polygons with the same area. A Dudeney dissection of α to β is a partition of α into parts which can be reassembled to produce β as follows: Hinge the parts of α like a string along the perimeter of α, then fix one of the parts to form β with the perimeter of α going into its interior and with its perimeter consisting of the dissection lines in the interior of α, without turning the surfaces over. In this paper we discuss a special case of Dudeney dissection where α is congruent to β, in particular, when all hinge points are on the vertices of the polygon α. We determine necessary and sufficient conditions under which such dissections exist.